{"id":19820,"date":"2026-02-01T18:14:27","date_gmt":"2026-02-01T10:14:27","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=19820"},"modified":"2026-02-01T18:14:27","modified_gmt":"2026-02-01T10:14:27","slug":"my-linear-algebra-learning-01-2","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2026\/02\/my-linear-algebra-learning-01-2\/","title":{"rendered":"\u7ebf\u6027\u4ee3\u6570\u56de\u987e(\u4e8c)"},"content":{"rendered":"\n\n\n\n<p style=\"margin-top:4px\">\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u524d\u9762\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\uff0c\u8fd8\u6709\u4e00\u4e9b\u95ee\u9898\u6ca1\u6709\u5f7b\u5e95\u56de\u7b54\uff08\u4f8b\u5982\uff0c\u89e3\u7a7a\u95f4\u7684\u63cf\u8ff0\uff09\uff0c\u5728\u56de\u7b54\u8fd9\u4e2a\u95ee\u9898\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u4e86\u89e3\u4e00\u4e0b\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u3002\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u7684\u4e25\u683c\u5b9a\u4e49\u662f\u6709\u4e9b\u67af\u71e5\u7684\uff0c\u8fd9\u91cc\u6682\u65f6\u628a\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u7684\u9650\u5236\u4e3a\u5927\u5bb6\u6240\u719f\u6089\u7684\u3001\u6700\u4e3a\u5178\u578b\u7684\u201c \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\u201d\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u5411\u91cf\u7a7a\u95f4<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">1.1 \u57fa<\/h4>\n\n\n\n<p>\u8981\u63cf\u8ff0\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u5143\u7d20\uff0c\u5219\u9996\u5148\u9700\u8981\u4e00\u7ec4\u57fa\uff08\u5750\u6807\uff09\u3002\u5728\\(n\\)\u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u6700\u4e3a\u5e38\u89c1\u7684\u4e00\u7ec4\u57fa\uff0c\u5373\u4e3a\u591a\u4e2a\u201c\u5782\u76f4\u201d\uff08\u201c\u6b63\u4ea4\u201d\uff09\u7684\u5355\u4f4d\u5411\u91cf\uff0c\u5373\uff1a<\/p>\n\n\n<p>$$ \\begin{align}<br \/>\n\\alpha_1 &#038;= (1,0,\\dots) \\\\<br \/>\n&#038; \\vdots \\\\<br \/>\n\\alpha_i &#038;= (\\dots,0,1,0,\\dots) \\\\<br \/>\n&#038; \\vdots \\\\<br \/>\n\\alpha_n &#038;= (0,\\dots,1)<br \/>\n\\end{align}<br \/>\n$$<\/p>\n\n\n\n<p>\u5728\u4e8c\u7ef4\u5e73\u9762\u7a7a\u95f4\u4e2d\uff0c\u5219\u4e3a\uff1a\\( \\alpha_1 = (1,0) \\quad \\alpha_2 = (0,1)\\)\uff1b\u4e09\u7ef4\u7a7a\u95f4\u5219\u4e3a\uff1a\\( \\alpha_1 = (1,0,0) \\quad \\alpha_2 = (0,1,0) \\quad \\alpha_3 = (0,0,1)\\)\u3002<\/p>\n\n\n\n<p>\u65e2\u7136\u6709\u201c\u6b63\u4ea4\u201d\u57fa\uff0c\u90a3\u4e48\u5f53\u7136\u6709\u4e0d\u90a3\u4e48\u201c\u6b63\u4ea4\u201d\u7684\u57fa\uff0c\u800c\u6b64\u7c7b\u201c\u57fa\u201d\u5219\u662f\u66f4\u4e3a\u666e\u904d\u7684\u3002\u4e8b\u5b9e\u4e0a\uff0c\u66f4\u4e3a\u666e\u904d\u7684\uff0c\u4efb\u4f55 \\( n \\) \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u90fd\u53ef\u4ee5\u4f5c\u4e3a\u5411\u91cf\u7a7a\u95f4\u7684\u57fa\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1.2 \u7ebf\u6027\u76f8\u5173\u4e0e\u7ebf\u6027\u65e0\u5173<\/h4>\n\n\n\n<p>\u8003\u8651\u4e00\u7ec4\u5411\u91cf\\( \\alpha_1,\\dots , \\alpha_n \\)\uff0c\u5982\u679c\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\\( a_i = 0 \\quad i=1,\\dots,n\\)\u65f6\u5982\u4e0b\u7684\u7b49\u5f0f\u624d\u6210\u7acb\uff1a<\/p>\n\n\n<p>$$ a_1\\alpha_1 + a_2\\alpha_2 + \\dots + a_n\\alpha_n = 0 $$<\/p>\n\n\n\n<p>\u90a3\u4e48\uff0c\u5c31\u8bf4 \u8fd9\u7ec4\u5411\u91cf \\( \\alpha_1,\\dots , \\alpha_n \\) \u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002\u53cd\u4e4b\uff0c\u5219\u79f0\u8fd9\u7ec4\u5411\u91cf\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002<\/p>\n\n\n\n<p>\u6216\u8005\u8fd9\u4e48\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u7ec4\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\\( \\alpha_1,\\dots , \\alpha_n \\)\uff1a\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf\u90fd\u4e0d\u80fd\u7528\u5269\u4f59\u7684\u5411\u91cf\u505a\u201c\u7ebf\u6027\u8868\u793a\u201d\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1.3 \u4e00\u4e9b\u91cd\u8981\u7684\u7ed3\u8bba<\/h4>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\u8bbe\\( \\{ \\alpha_1,\\dots , \\alpha_n \\}\\)\u662f\u5411\u91cf\u7a7a\u95f4\\( V \\)\u7684\u4e00\u7ec4\u57fa\uff0c\u90a3\u4e48\\( V \\)\u7a7a\u95f4\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf\u90fd\u53ef\u4ee5<strong>\u552f\u4e00<\/strong>\u7684\u8868\u793a\u4e3a\u8fd9\u7ec4\u57fa\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u4e2a\u7ebf\u6027\u7ec4\u5408\u7684\u7cfb\u6570\uff0c\u5c31\u53eb\u201c\u5750\u6807\u201d\uff08\u6ce8\uff1a\u76f8\u5bf9\u4e8e\u8fd9\u7ec4\u57fa\uff09\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\\( W_1 \\)\u3001\\( W_2 \\)\u662f\\( V \\)\u7684\u6709\u9650\u5b50\u7a7a\u95f4\uff0c\u90a3\u4e48\u6709\uff1a<\/p>\n\n\n<p>$$  dim(W_1+W_2) = dim(W_1) + dim(W_2) &#8211;  dim(W_1\\cap W_2)  $$<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\\( n \\)\u7ef4\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u4efb\u610f\\( n \\)\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u90fd\u53ef\u4ee5\u53d6\u505a\u57fa\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u7ebf\u6027\u53d8\u6362\u4e0e\u77e9\u9635<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">2.1 \u6982\u8ff0<\/h4>\n\n\n\n<p>\u201c\u7ebf\u6027\u53d8\u6362\u201d\u662f\u6307\u5411\u91cf\u7a7a\u95f4\u4e2d\u4e00\u7c7b\u7279\u6b8a\u7684\u6620\u5c04 \\( \\sigma : R^n \\to R^m \\) \uff0c\u9700\u8981\u6ee1\u8db3\u6761\u4ef6\u662f\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\( \\sigma(\\xi + \\eta)  = \\sigma(\\xi) + \\sigma(\\eta) \\)<\/li>\n\n\n\n<li>\\( \\sigma(a\\xi)  = a\\sigma(\\xi) \\)<\/li>\n<\/ul>\n\n\n\n<p>\u201c\u7ebf\u6027\u53d8\u6362\u201d \u63cf\u8ff0\u4e86\u5411\u91cf\u7a7a\u95f4\u4e4b\u95f4\u7684\u6620\u5c04\u3002\u540e\u7eed\u6240\u6709\u7684\u5185\u5bb9\u5927\u6982\u90fd\u662f\u56f4\u7ed5\u6b64\u800c\u5c55\u5f00\uff0c\u540e\u7eed\u6240\u6709\u7684\u5185\u5bb9\u90fd\u4f1a\u5c1d\u8bd5\u901a\u8fc7\u5404\u79cd\u65b9\u5f0f\u5c06 \u201c\u7ebf\u6027\u53d8\u6362\u201d \u7684\u7279\u6027\u7814\u7a76\u6e05\u695a\u3002\u8fd9\u91cc\u5199\u51fa\u90e8\u5206\u7ed3\u8bba\uff0c\u540e\u9762\u518d\u6162\u6162\u5c55\u5f00\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7ebf\u6027\u53d8\u6362\u4e4b\u4e0b\uff0c\u539f\u70b9\u4fdd\u6301\u4e0d\u53d8\u3002\u5373 \\( \\sigma( \\vec{0} ) = \\vec{0}  \\)<\/li>\n\n\n\n<li>\u51e0\u4f55\u610f\u4e49\u4e0b\uff0c\u901a\u5e38\uff0c\u7ebf\u6027\u53d8\u6362\u5305\u62ec\u4e86\uff1a\u65cb\u8f6c\u3001\u955c\u50cf\u3001\u62c9\u4f38\/\u538b\u7f29\uff08\u7279\u522b\u7684\uff0c\u6709\u65f6\u5019\u4f1a\u538b\u7f29\u5230\u96f6\uff09\u3001\u526a\u5207<\/li>\n<\/ul>\n\n\n\n<p>\u4e3a\u4e86\u7814\u7a76\u6e05\u695a\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4e0a\u8ff0\u7684\u7279\u70b9\uff0c\u901a\u5e38\u9700\u8981\u9009\u53d6\u4e00\u7ec4\u201c\u57fa\u201d\uff0c\u7136\u540e\u4f7f\u7528\u8fd9\u7ec4\u201c\u57fa\u201d\u7684\u201c\u5750\u6807\u201d\u6765\u63cf\u8ff0\u7a7a\u95f4\u4e2d\u7684\u70b9\uff0c\u8fdb\u800c\u518d\u63cf\u8ff0\u5bf9\u5e94\u7684\u7ebf\u6027\u53d8\u6362\u3002\u6700\u4e3a\u5e38\u89c1\u7684\u57fa\u4e3a\u201c\u6b63\u4ea4\u5355\u4f4d\u57fa\u201d\u3002<\/p>\n\n\n\n<p>\u4ece\u65b9\u6cd5\u4e0a\u6765\u770b\uff0c\u7814\u7a76\u6e05\u695a\u201c\u7ebf\u6027\u53d8\u6362\u201d\u6700\u4e3a\u5173\u952e\u7684\u662f\u7814\u7a76\u6e05\u695a\u5bf9\u5e94\u7684\u201c\u53d8\u6362\u77e9\u9635\u201d\u3002\u6240\u4ee5\uff0c\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u7684\u6838\u5fc3\u540e\u9762\u5c31\u53d8\u6210\u4e86\u5bf9\u77e9\u9635\u7279\u6027\u7684\u7814\u7a76\uff0c\u4f46\u662f\uff0c\u4e5f\u4e0d\u8981\u5fd8\u8bb0\u4e86\u521d\u8877\uff0c\u5426\u5219\u5f88\u5feb\u5c31\u8ff7\u5931\u4e86\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2.2 \u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n<p>\u6211\u5728\u5927\u5b66\u671f\u95f4\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362\u3001\u77e9\u9635\u6709\u4ec0\u4e48\u4f5c\u7528\uff0c\u662f\u5b8c\u5168\u6ca1\u6709\u6982\u5ff5\u7684\u3002\u6240\u4ee5\u5bf9\u4e8e\u4ed6\u4eec\u7684\u7279\u6027\u7814\u7a76\u4e5f\u6ca1\u6709\u638c\u63e1\u7684\u5f88\u6df1\uff0c\u57fa\u672c\u4e0a\u662f\u505c\u7559\u5728\u80fd\u591f\u628a\u4e00\u4e9b\u8054\u7cfb\u9898\u505a\u5bf9\u8fd9\u4e2a\u5c42\u9762\u3002\u800c\u73b0\u5728\uff0c\u6ce8\u610f\u7ebf\u6027\u53d8\u6362\u7684\u5e7f\u6cdb\u5e94\u7528\u4e4b\u540e\uff0c\u5c1d\u8bd5\u53bb\u7406\u89e3\u53bb\u672c\u8d28\u4e4b\u540e\uff0c\u5c31\u4f1a\u5bfb\u6839\u95ee\u5e95\u7684\u53bb\u7406\u89e3\u6e05\u695a\u4ec0\u4e48\u662f\u7ebf\u6027\u53d8\u6362\u3001\u4ec0\u4e48\u662f\u77e9\u9635\u3002\u8fd9\u91cc\u518d\u6b21\u8bf4\u8bf4\u6211\u7684\u7406\u89e3\u3002<\/p>\n\n\n\n<p>\u5728\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u6700\u4e3a\u5e38\u89c1\u7684\u662f \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u4f1a\u6709\u5f88\u591a\u7684\u5411\u91cf\uff0c\u4f8b\u5982\u6bcf\u4e2a Embedding \u5c31\u53ef\u4ee5\u7406\u89e3\u662f\u5728\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\u4e2d\uff0c<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u201c\u7ebf\u6027\u53d8\u6362\u201d\u8868\u8ff0<\/mark>\u4e86\u662f\u7a7a\u95f4\u4e2d\u7684\u4e00\u7c7b\u6620\u5c04\uff0c\u8be5\u6620\u5c04\u6ee1\u8db3\u4e0a\u8ff0\u201c\u5c0f\u7ed32.1\u201d\u4e2d\u7684\u4e24\u4e2a\u8981\u6c42\uff0c\u5373\u539f\u70b9\u4f9d\u65e7\u5230\u539f\u70b9\u3001\u6620\u5c04\u4fdd\u6301\u6240\u8c13\u7684\u201c\u7ebf\u6027\u201d\uff08\u4f8b\u5982\uff0c\u5411\u91cf\u548c\u7684\u6620\u5c04\u7b49\u4e8e\u6620\u5c04\u7684\u548c\u7b49\uff09\u3002<\/p>\n\n\n\n<p>\u5728\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5c31\u662f\u4e00\u4e2a\u7b26\u5408\u4e00\u5b9a\u6761\u4ef6\u7684\u6620\u5c04\u3002\u4e0e\u5411\u91cf\u7a7a\u95f4\u7684\u57fa\u7684\u9009\u53d6\u662f\u6ca1\u6709\u5173\u7cfb\u7684\u3002\u81ea\u6b64\uff0c\u4e0e\u77e9\u9635\u4e5f\u662f\u6ca1\u6709\u5173\u7cfb\u7684\u3002\u6240\u4ee5\uff0c\u7ebf\u6027\u53d8\u6362\u672c\u8eab\u662f\u66f4\u4e3a\u5e95\u5c42\u3001\u66f4\u4e3a\u57fa\u7840\u7684\u6982\u5ff5\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2.3 \u6b27\u6c0f\u7a7a\u95f4\u7684\u7ebf\u6027\u53d8\u6362\u4e0e\u77e9\u9635<\/h4>\n\n\n\n<p>\u73b0\u5728\u6211\u4eec\u628a\u95ee\u9898\u9650\u5b9a\u5728 \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u3002\u90a3\u4e48\uff0c\u8fd9\u65f6\u5019\uff0c\u6211\u4eec\u5982\u4f55\u63cf\u8ff0\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5462\uff1f\u662f\u7684\uff0c\u5c31\u662f\u201c\u57fa\u201d\u4e0e\u201c\u77e9\u9635\u201d\u3002<\/p>\n\n\n\n<p>\u901a\u5e38\uff0c\\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u6211\u4eec\u4f1a\u5148\u9009\u53d6\u4e00\u7ec4\u57fa\uff0c\u7136\u540e\u4f7f\u7528\u4e00\u4e2a\u77e9\u9635\u53bb\u63cf\u8ff0\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u3002\u5e76\u4e14\u975e\u5e38\u5e78\u8fd0\u7684\uff0c\u4e00\u65e6\u8fd9\u7ec4\u57fa \u9009\u5b9a\u4e86\uff0c\u8fd9\u4e2a\u77e9\u9635\u662f\u552f\u4e00\u7684\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\u5728\\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff08\u8fd9\u4e2a\u6761\u4ef6\u4f3c\u4e4e\u53ef\u4ee5\u53bb\u6389\uff09\u4e2d\uff0c\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362 \\(\\sigma \\)\uff0c\u5982\u679c\u9009\u5b9a\u4e00\u7ec4\u201c\u57fa\u201d\uff0c\u90a3\u4e48\u5c31\u5b58\u5728\u552f\u4e00\u7684\u201c\u77e9\u9635\u201d\u63cf\u8ff0\u8be5\u7ebf\u6027\u53d8\u6362\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u7ed3\u8bba\uff0c\u662f\u6bd4\u8f83\u660e\u663e\u7684\u3002\u6211\u4eec\u8003\u8651\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362\u4e2d\u7684\u4e0a\u8ff0\u9009\u5b9a\u7684\u57fa\u5411\u91cf \\(\\alpha_i,\\quad \\text{where } i = 1,\\ldots,n \\)\uff0c\u7ebf\u6027\u53d8\u6362\u5c06\u5176\u6620\u5c04\u5230 \\(\\beta_i,\\quad \\text{where } i = 1,\\ldots,n \\)\uff0c\u90a3\u4e48\u6839\u636e\u201c\u57fa\u201d\u7684\u57fa\u672c\u6027\u8d28\uff0c\u5bf9\u4e8e\u8fd9\u91cc\u7684\u4efb\u4f55 \\(\\beta_i \\)\u90fd\u53ef\u4ee5\u8868\u793a\u6210\\(\\alpha_i \\)\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u6240\u6709\u7684\u8fd9\u4e9b\u7cfb\u6570\u6784\u6210\u7684\u77e9\u9635\uff0c\u5c31\u662f\u4e0a\u8ff0\u63cf\u8ff0\u7684\u552f\u4e00\u7684\u201c\u77e9\u9635\u201d\u3002\u5177\u4f53\u7684\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\beta_1 &#038;= a_{11}\\alpha_1 + a_{21}\\alpha_2 + \\cdots + a_{n1}\\alpha_n \\\\[0.5em]<br \/>\n\\beta_2 &#038;= a_{12}\\alpha_1 + a_{22}\\alpha_2 + \\cdots + a_{n2}\\alpha_n \\\\[0.5em]<br \/>\n&#038;\\ \\vdots \\\\[0.5em]<br \/>\n\\beta_n &#038;= a_{1n}\\alpha_1 + a_{2n}\\alpha_2 + \\cdots + a_{nn}\\alpha_n<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5373\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\begin{bmatrix}<br \/>\n\\beta_1 &#038; \\beta_2 &#038; \\cdots &#038; \\beta_n<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n\\alpha_1 &#038; \\alpha_2 &#038; \\cdots &#038; \\alpha_n<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1n} \\\\<br \/>\na_{21} &#038; a_{22} &#038; \\cdots &#038; a_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nn}<br \/>\n\\end{bmatrix}<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5373\u5728\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\\(\\sigma \\) \u5728\u9009\u5b9a\u57fa \\(\\alpha_i,\\quad \\text{where } i = 1,\\ldots,n \\) \u5bf9\u5e94\u7684\u77e9\u9635\u4e3a\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nA =<br \/>\n\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1n} \\\\<br \/>\na_{21} &#038; a_{22} &#038; \\cdots &#038; a_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nn}<br \/>\n\\end{bmatrix}<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u4ece\u8fd9\u6bb5\u7b80\u6d01\u7684\u201c\u8bc1\u660e\u201d\uff08\u6216\u201c\u8bf4\u660e\u201d\uff09\u6765\u770b\uff0c\u6211\u4eec\u5f88\u81ea\u7136\u6709\u5982\u4e0b\u7ed3\u8bba\uff0c\u6839\u636e\u7a7a\u95f4\u4e2d\u7684\u201c\u57fa\u201d\u7684\u9009\u53d6\u4e0d\u540c\uff0c\u6211\u4eec\u4f1a\u5f97\u5230\u4e0d\u540c\u7684\u77e9\u9635\u3002\u56e0\u4e3a\u6211\u4eec\u53cd\u590d\u4f1a\u63d0\u5230\uff0c\u6211\u4eec\u7ecf\u5e38\u4f1a\u901a\u8fc7\u7814\u7a76\u77e9\u9635\u7684\u7279\u6027\u6765\u7814\u7a76\u7ebf\u6027\u53d8\u6362\u3002\u90a3\u4e48\uff0c\u540c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5728\u4e0d\u540c\u7684\u201c\u57fa\u201d\u4e0b\u7684\u4e0d\u540c\u201c\u77e9\u9635\u201d\uff0c\u5f88\u81ea\u7136\u7684\u80fd\u591f\u60f3\u5230\uff0c\u8fd9\u4e9b\u201c\u77e9\u9635\u201d\u662f\u6709\u67d0\u4e9b\u5171\u6027\u7684\uff0c\u662f\u7684\uff0c\u6211\u4eec\u79f0\u8fd9\u4e9b\u77e9\u9635\u4e3a\u201c\u76f8\u4f3c\u77e9\u9635\u201d\uff0c\u76f8\u5173\u7279\u6027\uff0c\u6682\u4e0d\u5c55\u5f00\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. \u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">3.1 \u4e3a\u4ec0\u4e48<\/h4>\n\n\n\n<p>\u4e3a\u4ec0\u4e48\u6211\u4eec\u9700\u8981\u5173\u6ce8\u201c\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\u201d\u5462\uff1f\u4e3a\u4ec0\u4e48\u6211\u4eec\u8981\u53bb\u4e86\u89e3\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u5462\uff1f<\/p>\n\n\n\n<p>\u539f\u56e0\u662f\u201c\u7ebf\u6027\u53d8\u6362\u201d\u662f\u4e00\u4e2a\u6620\u5c04\uff0c\u662f\u975e\u5e38\u62bd\u8c61\u7684\u3002\u800c\u7279\u5f81\u503c \u3001\u7279\u5f81\u5411\u91cf\u3001SVD\u5206\u89e3\u53ef\u4ee5\u628a\u7ebf\u6027\u53d8\u6362\u6700\u4e3a\u5173\u952e\u7684\u7279\u6027\uff0c\u4ee5\u975e\u5e38\u201c\u76f4\u89c2\u201d\u7684\u5f62\u5f0f\u8868\u8fbe\u51fa\u6765\u3002\u5f53\u7136\uff0c\u8fd9\u91cc\u7684\u201c\u76f4\u89c2\u201d\u5e76\u4e0d\u662f\u7b80\u5355\u610f\u4e49\u4e0a\u80fd\u591f\u4e00\u773c\u5c31\u770b\u51fa\u4ec0\u4e48\u6765\uff0c\u4e8b\u5b9e\u4e0a\uff0c\u201c\u7ebf\u6027\u53d8\u6362\u201d\u672c\u8eab\u5c31\u6709\u5f88\u5f3a\u7684\u62bd\u8c61\u6027\uff0c\u8fd9\u91cc\u7684\u201c\u76f4\u89c2\u201d\u53ea\u662f\u76f8\u5bf9\u7684\uff0c\u662f\u5426\u76f4\u89c2\uff0c\u5b8c\u5168\u4f9d\u8d56\u4e8e\u5404\u4f4d\u770b\u5ba2\u81ea\u5df1\u7684\u201c\u609f\u6027\u201d\u4e86 \u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7279\u5f81\u5411\u91cf\u7684\u57fa\u672c\u5b9a\u4e49\uff1a<\/mark>\u5982\u679c\u6709 \\( \\sigma(\\xi) = \\lambda \\xi \\) \uff0c\u90a3\u4e48\u8fd9\u91cc\u7684  \\( \\xi \\) \u5c31\u662f\u7279\u5f81\u5411\u91cf\uff0c\u5bf9\u5e94\u7684 \\(\\lambda \\) \u5c31\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n<p>\u8981\u60f3\u771f\u6b63\u8bf4\u6e05\u695a\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u662f\u9700\u8981\u975e\u5e38\u591a\u7bc7\u5e45\u7684\uff0c\u800c\u4e14\u5173\u4e8e\u5bf9\u7279\u5f81\u5411\u91cf\u7684\u7406\u89e3\u5bf9\u4e8e\u7406\u89e3\u7ebf\u6027\u53d8\u5316\u4e5f\u662f\u975e\u5e38\u5173\u952e\u7684\uff0c\u6240\u4ee5\uff0c\u5efa\u8bae\u82b1\u4e9b\u65f6\u95f4\u8f83\u4e3a\u7cfb\u7edf\u7684\u505a\u4e00\u4e9b\u7406\u89e3\u3002\u5982\u679c\u4f60\u5df2\u7ecf\u5efa\u7acb\u7684\u57fa\u7840\u6982\u5ff5\uff0c\u8fd9\u91cc\u7684\u4e00\u7bc7\u6587\u7ae0\u53ef\u80fd\u662f\u5e2e\u52a9\u4f60\u589e\u5f3a\u4e00\u4e9b\u7406\u89e3\uff1a\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3.2 \u5173\u4e8e\u5bf9\u7279\u5f81\u5411\u91cf\u7684\u7406\u89e3<\/h4>\n\n\n\n<p>\u5b8c\u6574\u7684\u8ba8\u8bba\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u662f\u590d\u6742\u7684\uff0c\u8fd9\u91cc\u5c06\u5176\u9650\u5b9a\u5728\u4e00\u4e9b\u8f83\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u5165\u95e8\u3002\u6211\u4eec\u8fd9\u91cc\u8003\u8651\u6700\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u5373\u5bf9\u4e8e\u4e00\u4e2a \\(n \\times n \\)\u7684\u77e9\u9635\uff0c\u5176\u79e9\u4e3a \\(n \\)\uff0c\u5e76\u4e14\u5728\u8ba1\u7b97\u7279\u5f81\u503c\u65f6\uff0c\u6709 \\(n \\) \u662f\u4e0d\u91cd\u590d\u7684\u5b9e\u6570\u89e3\uff0c\u5373\u6ca1\u6709\u4efb\u4f55\u6839\u5f0f\u91cd\u6839\u3002\u5982\u679c\uff0c\u6070\u597d \\( n = 2 \\)\u8fd9\u5927\u6982\u662f\u6700\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\u4e86\uff0c\u4e0d\u8fc7\u7406\u89e3\u8fd9\u79cd\u60c5\u51b5\uff0c\u518d\u8fdb\u4e00\u6b65\u62d3\u5c55\uff0c\u5219\u5b66\u4e60\u66f2\u7ebf\u4f1a\u5e73\u6ed1\u5f88\u591a\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u6765\u770b\u4e00\u4e2a\u5b9e\u4f8b\uff0c\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u5728\u6807\u51c6\u57fa\u4e0b\uff0c\u6211\u4eec\u6709\u5982\u4e0b\u7684\u7ebf\u6027\u53d8\u6362\u77e9\u9635\uff1a<\/p>\n\n\n<p>$$<br \/>\nW = \\begin{bmatrix}<br \/>\n2 &#038; 1 \\\\<br \/>\n1 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n$$<\/p>\n\n\n\n<p>\u6839\u636e\u4e0a\u8ff0\u7279\u5f81\u5411\u91cf\u7279\u5f81\u503c\u7684\u5b9a\u4e49\u8fdb\u884c\u6c42\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u6709\u5982\u4e0b\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\( \\lambda_1 = 3 \\) \u7279\u5f81\u5411\u91cf \\( (1,1) \\)<\/li>\n\n\n\n<li>\\( \\lambda_2 = 1 \\) \u7279\u6027\u5411\u91cf \\( (-1,1) \\)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image alignright size-full is-resized\" style=\"margin-right:var(--wp--preset--spacing--small)\"><img loading=\"lazy\" decoding=\"async\" width=\"694\" height=\"682\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18.png\" alt=\"\" class=\"wp-image-22851\" style=\"width:316px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18.png 694w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18-300x295.png 300w\" sizes=\"auto, (max-width: 694px) 100vw, 694px\" \/><\/figure>\n\n\n\n<p>\u4ece\u7279\u5f81\u5411\u91cf\u89d2\u5ea6\u7406\u89e3\u7ebf\u6027\u53d8\u6362\uff1a\u90a3\u4e48\u4e0a\u8ff0\u7684\u77e9\u9635A\u5bf9\u5e94\u7684\u7ebf\u6027\u53d8\u6362 \\(\\sigma \\) \u6709\u5982\u4e0b\u7279\u6027\uff0c\u5728\u8fd9\u4e2a\u4e8c\u7ef4\u7a7a\u95f4\u4efb\u4f55\u5411\u91cf \\(\\beta \\)\uff0c\u90fd\u53ef\u4ee5\u5206\u89e3\uff08\u6295\u5f71\uff09\u4e3a\u4e0a\u8ff0\u4e24\u4e2a\u7279\u5f81\u5411\u91cf\u65b9\u5411\u7684\u5411\u91cf\uff1a \\(\\beta_1 \\,, \\beta_2 \\)\uff0c\u4e14\u6709\uff1a\\(\\beta = \\beta_1 + \\beta_2 \\)\u3002\u90a3\u4e48\uff0c\u5219\u6709\uff1a\\(\\sigma(\\beta) = \\lambda_1  \\beta_1 + \\lambda_2  \\beta_2  \\)\u3002\u5373\uff0c\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u53ef\u4ee5\u8fd9\u6837\u63cf\u8ff0\uff1a\u5148\u5c06\u4efb\u4f55\u5411\u91cf\u6cbf\u7740\u7279\u5f81\u5411\u91cf\u65b9\u5411\u5206\u89e3\uff0c\u7136\u540e\u518d\u6309\u7167\u7279\u5f81\u503c\u7684\u5927\u5c0f\u8fdb\u884c\u62c9\u4f38\u6216\u538b\u7f29\uff0c\u7136\u540e\u518d\u628a\u5411\u91cf\u5408\u5e76\u8d77\u6765\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u89e3\u91ca\uff0c\u53ef\u4ee5\u5bf9\u7167\u7740\u53f3\u56fe\u53bb\u7406\u89e3\u3002\u7279\u5f81\u5411\u91cf\u5206\u522b\u4e3a \\((1,1) \\) \u548c \\((-1,1) \\) \uff0c\u5373\u56fe\u4e2d\u6d45\u7eff\u8272\u3001\u6d45\u84dd\u8272\u65b9\u5411\u3002\u8be5\u77e9\u9635\u4f5c\u7528\u5728\u5411\u91cf \\((0,1) \\) 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[&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":22817,"comment_status":"open","ping_status":"closed","sticky":false,"template":"wp-custom-template-a-1440-px-width-template","format":"standard","meta":{"_eb_attr":"","inline_featured_image":false,"_tocer_settings":[],"footnotes":""},"categories":[1],"tags":[],"class_list":["post-19820","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-simplelife"],"_links":{"self":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/19820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/comments?post=19820"}],"version-history":[{"count":66,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/19820\/revisions"}],"predecessor-version":[{"id":23221,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/19820\/revisions\/23221"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media\/22817"}],"wp:attachment":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media?parent=19820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/categories?post=19820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/tags?post=19820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}